Approximating Common Fixed Points of an Evolution Family on a Metric Space via Mann Iteration

Luo, Liang; Ullah, Rizwan; Rahmat, Gul; Butt, Saad Ihsan; Numan, Muhammad

doi:https://doi.org/10.1155/2021/6764280

Journal of Mathematics

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6764280 | https://doi.org/10.1155/2021/6764280

Approximating Common Fixed Points of an Evolution Family on a Metric Space via Mann Iteration

Liang Luo,^1,2Rizwan Ullah,³Gul Rahmat,³Saad Ihsan Butt,⁴and Muhammad Numan⁵

Academic Editor: Ming-Sheng Liu

Received04 Mar 2020

Revised20 Aug 2020

Accepted05 Mar 2021

Published25 Mar 2021

Abstract

In this article, we present the set of all common fixed points of a subfamily of an evolution family in terms of intersection of all common fixed points of only two operators from the family; that is, for subset of , we have , where and are positive and is an irrational number. Furthermore, we approximate such common fixed points by using the modified Mann iteration process. In fact, we are generalizing the results from a semigroup of operators to evolution families of operators on a metric space.

1. Introduction

The theory of fixed points has a lot of applications in mathematics. The existence of a solution for a system is equivalent to the existence of a fixed point for a particular map associated with the system. Fixed point theory helps us for assurance of existence and uniqueness of solutions of differential equations, integral equations, algebraic system, and numerical equations. For further study of fixed point theory and applications, we refer to [1]. Since 1960, an extensive amount of work has been done in the field of fixed point theory. Belluce and Kirk [2, 3], Browder and Bruck [4, 5], Lim [6], and DeMarr [7] have discussed fixed point for contractions and nonexpansive mappings on Banach spaces. However, in [8], the occurring of common fixed points of nonexpansive strongly continuous semigroups has been discussed via employing asymptotic type approach.

A family of bounded linear mappings is called a semigroup if it satisfies the following two conditions: and , for all and . Such families arise from the solution of the autonomous system:

The study of semigroups has a big role in the field of difference and differential equations. Although it is very important, still cannot be applied for the nonautonomous system:

For such a system, we need the idea of evolution family. A family of bounded linear mappings on a metric space is said to be an evolution family if it satisfies the following two conditions:(i)(ii) for all

Such families are the generalized form of a semigroup.

Therefore, in our study, we investigate the common fixed points of an evolution family (instead of semigroup) for a hyperbolic metric space. For some recent published work on this topic, we refer to [9–11]. In fact, in [9], the following result is proved.

Theorem 1. Let be a subfamily of an evolution family of nonexpansive and strongly continuous operators on a metric space . If are two positive real numbers with restriction that is an irrational number, thenwhere AFPS mean approximate fixed point sequence of the operators.

In [9], authors discussed the approximate fixed point sequences, but this paper is about convergence to a common fixed point of an evolution family by using the Mann iteration process. Also, we present the set of all common fixed points by the intersection of all common fixed points of only two operators from the same family. Therefore, we mention that these are two different approaches for fixed points.

In [12], Khamsi and Buthinah discussed the fixed points of a semigroup on a metric space. Khamsi and Kozlowski [13, 14] presented fixed points for pointwise nonexpansive and pointwise contraction mapping in modular function spaces. Furthermore, the existence of fixed points for asymptotic and pointwise contractions and asymptotic and pointwise nonexpansive mappings has been discussed by Kirk and Xu in [15]. These results have been extended by Hussain and Khamsi [16] to metric spaces. In [17], Khamsi discussed the existence of nonlinear semigroups and gave some applications to differential equations in Musielak–Orlicz spaces.

2. Definitions and Preliminary Results

In this section, we present some definition which will be utilized in main results of this paper. Throughout the paper, we use the following notations for simplifications and to avoid repetitions: : denotes the set of real, positive real, irrational, rational, and integer numbers, respectively : denotes a metric space : denotes a hyperbolic metric space : with the restriction that

Definition 1 (see [18]). is said to be convex metric space if for all in , there exist a unique metric segment , and for any two points and , such that , and the following holds:where is the set of all line segments between the points of .

Definition 2 (see [19]). A convex is called hyperbolic metric space if for all in and , and the following condition holds:A subset of is convex if for all , in , and the line segment belongs to .

Definition 3 (see [12]). A is uniformly convex if for all , and for all positive real numbers and , the following inequalities hold:The concept of convexity in a uniform sense was developed initially only for Banach spaces (see [20]), but latter on in [21], it was extended to metric spaces. For further details, see [19, 22, 23].

Definition 4 (see [12]). A is called strictly convex when we havefor any and , and then .

Lemma 1 (see [14, 23]). Let be uniformly convex. Also, let in be a sequence of nonempty, nonincreasing, convex, closed, and bounded sets if and

If is with , then is a Cauchy sequence.

In , if there exist any nonincreasing sequence of sets which are nonempty, convex, closed, bounded, and also their intersection is nonempty, such property is called the property . Moreover is called complete if it has the property . Similarly if is complete and uniformly convex, then it satisfies the property ; for more details, we refer to [14, 23, 24].

Definition 5 (see [12]). Let is a nonempty subset of and be a semigroup on . The set of all common fixed points of the family is define as

Proposition 1 (see [12]). If is a nonempty and closed subset of a , the family be a semigroup of continuous operators. Moreover, if , then

Proposition 2 (see [12]). Let be a nonempty and closed subset of a and be a semigroup on . If , then there exist as defined in , such that

If we combine Propositions 1 and 2, we get following theorem.

Theorem 2 (see [12]). Let be a nonempty and closed subset of a , and be a semigroup. Then, for any two as defined in , we have

Definition 6. An operator is said to be nonexpansive if for any , the following inequality:holds. Moreover, is said to be fixed point of if for some and . We denote bywhich is the set of all fixed points of the operator .

Definition 7. An evolution family of bounded linear operators on a is called nonexpansive, continuous, and periodic if the following holds:(i), and (ii)(iii)There exist such that for all , respectivelyThe present paper deals with the study of exploring fixed points for a subfamily of evolution family. Let be a subfamily of the family on a metric space , that is, and periodic, where as defined in .
First, we define the set of all common fixed points of the above families and .The following lemma will be useful in the proof of our main results.

Lemma 2 (see [5]). Let which is strictly convex . If are nonexpansive operators such that , then for each and for nonexpansive mapping such that for , we have .

In the rest of the paper, we will denote the evolution family by , the subfamily of the evolution family by , and the nonempty closed and convex subset of the metric space unless stated otherwise by .

3. Methodology

In this paper, we provide results concerning the fixed points of a subfamily of a nonexpansive evolution family of operators. In fact, such results are presented in [12] for semigroups of nonexpansive bounded linear operators. So, in this paper, we use the methodology as used in [12]. Such results are new, and we do not find in the existing literature.

4. Main Results

In this section, we will present our main results concerning the fixed points of an evolution family of bounded linear operators in terms of intersection of sets of fixed points of two operators from the family.

Proposition 3. Let be an evolution family on . If such that , then

Proof. Clearly . We only need to prove thatLet which means for all , and . We need to show that . As , then by denseness property, there exist sequences such that and and . Now, consider for all .
Now taking limit on both sides, that is, for all gives . Hence, , . We get

Proposition 4 (see [25]). If is nonempty and additive subgroup, then either or for nonnegative number , we have . If there exist two real numbers as defined in , then the setis dense in . Moreover, if , then (W is dense in ).

Proposition 5. Let be subfamily of evolution family from into . If such that these numbers are also the periods of the evolution family, thenwhere .

Proof. As and are in , therefore . We need only to prove thatLet , then and . Let , then .
ConsiderHence, , where we use the property which is the identity-map and consider , where either or is negative. Assume that , thenHence, for any , and this completes the proof.

Theorem 3 (see [26]). Let be subfamily of the evolution family on . If as defined in , then

In particular, we have

Theorem 4. Let be subfamily of evolution family from into which is continuous and nonexpansive. If as defined in , thenwhere .

Proof. Let us consider . This ensures that and . Now,Hence, .
Again consider which implies or .
If we take and , it gives and , respectively. Hence, . This completes the proof.

5. Approximating of Fixed Points of an Evolution Family

In this section, we discuss the behavior of Mann iteration process by two operators, using previous results. With the support of these iterations, we will approximate the common fixed points of subfamily of an evolution family.

Definition 8. Let , which is nonexpansive mapping and . Also let represent the Mann iteration process produced by the constant and the mapping , which is given by following iteration formula:where is chosen arbitrarily in .

Lemma 3 (see [14, 23]). Let is uniformly convex metric space and such thatfor some . Then,

Lemma 4. Let is uniformly convex and be a nonexpansive mapping defined from to . Let and be a sequence defined by (27). Suppose . Then, exists.

Proof. Let us consider for all ,Since the sequence and is nonincreasing, it is bounded below, and hence it is convergent. Next, we define asymptotic radius of a sequence.

Definition 9 (see [12]). Let be a nonempty subset and be a bounded sequence in . Define byThe asymptotic radius of the sequence with respect to is denoted by and is defined by the following:where is used to represent the asymptotic radius of the sequence with respect to . Let such thatThen, is an asymptotic center of sequence with respect to .
Let represent the collection of all asymptotic centers of with respect to . When , we use the notation . Generally, the set of a bounded sequence contains more than one point. It can be empty too; for more details, we refer to [23].
Kuczumow [27] and Lim [6] offered an approach to weak convergence, which they called -convergence. In the case of spaces, this approach was very effective and successful.

Definition 10. Let be a bounded sequence in . This sequence is called -converge to if and only if for every subsequence of , is the unique asymptotic center of this subsequence. When is -converge to , we denote by .
Next, we discuss the iteration scheme (27) for nonexpansive mappings, and particularly, we will investigate the -convergence of hyperbolic spaces which are uniformly convex.

Lemma 5 (see [23, 28]). If is uniformly convex, then with respect to , there exist a unique asymptotic center for each bounded sequence .

Lemma 6 (see [23, 28]). If is uniformly convex and is a bounded sequence such that and and let is a sequence such that , then .

In uniformly convex Banach spaces, D. Gohde gave the demi-closed principle [29]; a similar concept is discussed in next theorem.

Lemma 7. Let be complete and uniformly convex and is a nonexpansive mapping on . If be a sequence such that and if is the asymptotic center of the sequence with respect to , then , in certain, if has approximate fixed point sequence , such that , then .

In order to study the iterative behavior defined by (27), we give the following necessary theorem.

Theorem 5 (see [28]). Let be uniformly convex and be as defined in Lemma 7, which has a fixed point. If , , and is generated by (27), thenand has an approximate fixed point sequence .

The condition that has a fixed point that can be removed if we assume that is bounded (see [23]). Now, we study the -convergence of the Mann iteration process given in (27).

Theorem 6. Let be uniformly convex and be as defined in Lemma 7, which has a fixed point. If , , and is generated by (27), then is -converge to .

Proof. The fixed set of operator is convex and closed, as given in [23]. Applying Theorem 5, has approximate fixed point sequence , which implies , if the sequence has a unique asymptotic center . Then, using Lemma 7, . Now, we prove that is -converge to . Suppose that the sequence has a subsequence and suppose this subsequence has the unique asymptotic center . Again since has an approximate fixed point sequence , we conclude that .
Hence,Since , we getSince has the unique asymptotic center , . This shows that -converge to .
The combination of Theorems 3 and 6 gives the result, which is discussed in the next theorem.

Theorem 7. Let is complete and uniformly convex with be a subset of . If is a subfamily of an evolution family on which is continuous and nonexpansive, if as defined in and are such that , and also if and be a sequence defined bywhere , then the sequence is -converge to a common fixed point of .

Proof. SetObserve that is nonexpansive. Clearly we have . Let and is the sequence produced by (37). Then,where . It means that the sequence is the same as produced by (27) for the operator . According to Theorem 6, -converge to some . By Theorem 3, we know that ; hence, which completes the proof.
We conclude this section by the following important remark.

Remark 1. If we put the condition of periodicity on the family for every positive real number, then it becomes a semigroup. So, in this case, Propositions 3.1, 3.2, and Theorems 3.2 and 4.3 in [12] become subcases of our work.

6. Conclusion

In this paper, we established the results concerning the fixed points of a subfamily of an evolution family on a metric space. It is more difficult to derive or find the set of all common fixed points for a family than to obtain the set of all common fixed points only for two maps. So, in this paper, we provide results which make it easy to present such set of fixed points in terms of only two maps from the same family. Also, we approximated the common fixed points for subfamily of evolution family using the well-known Mann iteration process. In future, we are trying to generalize these results for periodic evolution families which seem to be more applicable and fruitful for researchers working in this field or others related to fixed point theory. Recently, we completed more projects in this area, and we hope to be publish them in near future (see [30–32]).

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this paper.

Acknowledgments

This research was supported by the Natural Science Fund of Anhui Wenda University of Information Engineering (grant no. XZR2019A10).

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Copyright © 2021 Liang Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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